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Complete control Lyapunov functions: Stability under state constraints
Philipp Braun, Christopher Kellett, Luca Zaccarian
To cite this version:
Philipp Braun, Christopher Kellett, Luca Zaccarian. Complete control Lyapunov functions: Stability under state constraints. 11th IFAC Symposium on Nonlinear Control Systems, Sep 2019, Vienna, Austria. �hal-01995208�
Complete control Lyapunov functions:
Stability under state constraints?
Philipp Braun∗ Christopher M. Kellett∗ Luca Zaccarian∗∗
∗School of Electrical Engineering and Computing, University of Newcastle, Callaghan, New South Wales 2308, Australia, (e-mail:
{philipp.braun,chris.kellett}@newcastle.edu.au.
∗∗Laboratoire d’Analyse et d’Architecture des Syst`emes-Centre Nationnal de la Recherche Scientifique, Universit´e de Toulouse, CNRS,
UPS, 31400 Toulouse, France, and Dipartimento di Ingegneria Industriale, University of Trento, 38122 Trento, Italy,
luca.zaccarian@laas.fr.
Abstract:Lyapunov methods are one of the main tools to investigate local and global stability properties of dynamical systems. Even though Lyapunov methods have been studied and applied over many decades to unconstrained systems, extensions to systems with more complicated state constraints have been limited. This paper proposes an extension of classical control Lyapunov functions (CLFs) for differential inclusions by incorporating in particular bounded (nonconvex) state constraints in the form of obstacles in the CLF formulation. We show that the extended CLF formulation, which is called a complete CLF (CCLF) in the following, implies obstacle avoidance and weak stability (or asymptotic controllability) of the equilibrium of the dynamical system. Additionally, the necessity of nonsmooth CCLFs is highlighted. In the last part we construct CCLFs for linear systems, highlighting the difficulties of constructing such functions.
Keywords:(nonsmooth) control Lyapunov functions; asymptotic controllability; obstacle avoidance; nonsmooth controller design.
1. INTRODUCTION
Lyapunov functions (LFs) or control Lyapunov functions (CLFs) are a well studied tool to investigate stability and controllability properties of equilibria of dynamical systems with and without input. While necessary and sufficient conditions for the existence of LFs and CLFs characterizing stability properties of a specific equilibria globally or locally (i.e., by restricting the domain to a sublevel set of the LF/CLF) have been derived, only a few papers concentrate on the extension of classical Lyapunov theory to incorporate more complicated constraints in the Lyapunov formulation to guarantee stability properties and obstacle avoidance simultaneously. A possible reason is the lack of constructive methods to design CLFs for nonlinear systems even in the unconstrained setting.
In this paper we propose an extension of classical CLFs, which we call complete CLFs (CCLFs) and which in par- ticular allow us to consider bounded obstacles in the state space in the form of state constraints. We show that the existence of a CCLF guarantees asymptotic stabilizability of the origin for the dynamical system and simultaneous obstacle avoidance. Due to the topological obstruction of bounded obstacles in the state space, discontinuous feed- back laws and subsequently nonsmooth CLFs and CCLFs need to be considered to guarantee asymptotic stability and obstacle avoidance simultaneously (Liberzon, 2003, Chapter 4). Related approaches combining Lyapunov func-
? P. Braun and C. M. Kellett are supported by the Australian Research Council (Grant number: ARC-DP160102138).
tions with control barrier functions or methods using so- called artificial potential fields concentrate on smooth rep- resentations and consequently do not consider discontinu- ous feedback laws.
Control barrier functions in the context of dynamical sys- tems were introduced in Wieland and Allg¨ower (2007) as a certificate to ensure obstacle avoidance, or equivalently to ensure that a given set of unsafe states is never entered.
Since in general control barrier functions are assumed to be smooth, articles combining ideas of Lyapunov functions and control barrier functions (such as Ames et al. (2017), Ngo et al. (2005) or Tee et al. (2009)) cannot be used for controller design guaranteeing obstacle avoidance of bounded obstacles and asymptotic stability. Artificial po- tential fields in the robotics literature date back to the work in Khatib (1985), Khatib (1990), and with a recent relevant contribution in Paternain et al. (2018). Artifi- cial potential fields define smooth Lyapunov-like functions whose gradient is used to ensure obstacle avoidance and asymptotic stability. However, due to the assumption of a smooth function, it is well known and acknowledged in the literature that avoidance and stability can at best be guaranteed for sets excluding a set of measure zero in the state space but not for every initial value. Additionally, artificial potential fields are generally constructed for fully actuated systems. Underactuated systems where it is only possible to manipulate the dynamics in the direction of subspaces ofRn are only partially covered.
The contributions of this paper are as follows. In Section 2.2 nonsmooth CCLFs as an extension of CLFs are intro-
duced providing the possibility to incorporate (bounded) obstacles in the formulation of dynamical systems. Ad- ditionally, we show that the existence of a CCLF implies obstacle avoidance and asymptotic stability of the dynam- ical system before we discuss the necessity of nonsmooth CCLFs via several examples. In Section 3 we propose a possible form of a CCLF by combining several smooth functions. Additionally we show under which conditions this construction leads to a CCLF for linear systems but also highlight the difficulties in the design process.
We use the following notation. For x,xˆ ∈ Rn we define
|x| =pPn
i=1xni and |x|xˆ =|x−x|.ˆ Br(x) = {y ∈ Rn :
|y|x< r}, r >0, denotes an open ball andBr(x) denotes its closure.K,K∞ andKLare the standard notations for comparision functions andP denotes the class of positive definite functions (see Braun et al. (2018a), for example).
2. NONSMOOTH COMPLETE CONTROL LYAPUNOV FUNCTIONS
2.1 Mathematical setting
We consider differential inclusions
˙
x∈F(x) (1)
where F : Rn ⇒ Rn. An absolutely continuous function x:R≥0→Rn is a solution of the differential inclusion (1) from initial condition x(0) ∈ Rn if ˙x(t) ∈ F(x(t)) for almost all t ∈ R≥0. In a common abuse of notation, we usexandx(·) to denote points inRnand solutions to (1).
Additionally,S(x) denotes the set of solutions starting at x=x(0). To ensure existence of solutions of (1) we make the following standard assumptions.
Assumption 1. Consider the set-valued mapF :Rn ⇒Rn with 0∈F(0). We impose the following conditions onF: (i) (Regularity) F has nonempty, compact, and convex
values on Rn, and is upper semicontinuous, i.e., for all ε > 0 there exists a δ > 0 such that ξ ∈ Bδ(x) impliesF(ξ)⊂F(x) +Bε(0).
(ii) (Local boundedness) For each r > 0 there exists M >0 such that|x|< rimplies supw∈F(x)|w| ≤M. (iii) (Local Lipschitz continuity) For each x ∈ Rn\{0}
there exists a constant L > 0 and a neighborhood x∈ O ⊂Rn such thatF(x1)⊂F(x2) +BL|x1−x2|(0)
for allx1, x2∈ O. y
Assumption 1(i) ensures that the set of solutions S(x) is nonempty. Since we will discuss nonsmooth functions in the following, we use the (lower right) Dini derivative to extend the directional derivative to nondifferentiable functions. For a Lipschitz continuous functionϕ:Rn →R the Dini derivative in directionv∈Rn, is defined as
dϕ(x;v) = lim inf
t&0 1
t(ϕ(x+tv)−ϕ(x)).
If ϕ : Rn → R is continuously differentiable on a neigh- borhood containing x ∈Rn, then dϕ(x;v) = h∇ϕ(x), vi.
holds. A Lipschitz continuous function is continuously dif- ferentiable almost everywhere due to Rademacher’s theo- rem. It was shown in Sontag (1983) that weakKL-stability (phrased as an asymptotic controllability property of a controlled differential equation) is equivalent to the ex- istence of a continuous CLF. Subsequently, and indepen- dently, Rifford (2000) and Kellett and Teel (2000) (see also
Kellett and Teel (2004)) demonstrated the equivalence of weakKL-stability and the existence of a locally Lipschitz CLF.
Definition 1. The differential inclusion (1) is weaklyKL- stable with respect to the origin if there exists β ∈ KL such that, for eachx(0) =ξ∈Rn, there existsx(·)∈ S(ξ) so that|x(t)| ≤β(|ξ|, t) for allt≥0. y Theorem 1.(Nonsmooth CLFs). Suppose F satisfies As- sumption 1. Then the differential inclusion (1) is weakly KL-stable with respect to the origin if and only if there exists a Lipschitz continuous CLFV :Rn→R≥0,α1, α2∈ K∞, andρ∈ P such that
α1(|x|)≤V(x)≤α2(|x|), ∀x∈Rn (2) and for eachx∈Rn there existsw∈F(x) such that
dV(x;w)≤ −ρ(|x|). (3) 2.2 Complete control Lyapunov functions
In this subsection we extend Definition 1 to incorporate avoidance properties in the stability definition.
Definition 2. LetO ⊂Rn, 0∈ O, be open. The differential/ inclusion (1) is weaklyKL-stable with respect to the origin with avoidance property with respect toO, if there exists β ∈ KL such that, for each ξ ∈ Rn\O, there exists x(·)∈ S(ξ) so that
|x(t)| ≤β(|ξ|, t) and x(t)∈ O/ ∀t≥0.
y Though not stated explicitly here, we assume that O is nonempty. In the case whereO=∅, Definition 2 reduces to Definition 1. Since weakKL-stability can be equivalently concluded from the existence of a CLF, we consider an extension of CLFs appropriate for Definition 2.
Definition 3.(CCLF). Suppose that F satisfies Assump- tion 1. For i∈[1 :N], N ∈N, let Oi ⊂Rn be open sets and let VC : Rn →Rbe a Lipschitz continuous function.
Assume there existα1, α2∈ K∞and ρ∈ P such that the following properties are satisfied:
(i) For alli= [1 :N], there existci∈Rsuch that VC(x) =ci ∀x∈∂Oi and ci≤ min
x∈OiVC(x) (4) (ii) VC is positive definite and radially unbounded, i.e.,
α1(|x|)≤VC(x)≤α2(|x|). (5) (iii) For each x ∈ Rn\ ∪Ni=1Oi
there exists w ∈ F(x) such that
dVC(x;w))≤ −ρ(x). (6) ThenVCis called a Complete Control Lyapunov Function
(CCLF). y
Without loss of generality, we assume thatci >0 for all i∈[1 :N]. Ifci= 0 for some i∈[1 :N], then the radial unboundedness ofVC implies that ∂Oi={0}or ∂Oi=∅ and thus Oi = Rn\{0} or Oi = Rn. In both cases, the assumptions on the functions involved in Definition 2 and 3 can be trivially satisfied.
Theorem 2. Consider the differential inclusion (1) satisfy- ing Assumption 1. Additionally, letOi,i∈[1 :N],N ∈N, be open sets and letVC:Rn→Rbe a CCLF according to Definition 3. Then the differential inclusion (1) is weakly KL-stable with respect to the origin and has the avoidance property with respect toO=∪Ni=1Oi. y
The proof of Theorem 2 can be obtained by making minimal changes to the proofs of (Braun et al., 2018a, Thm. 4.11 and Thm. 4.12). For completeness, we report the proof of Theorem 2 in Appendix A. Theorem 2 provides the opportunity to incorporate constraints in the Lyapunov framework. In particular if the state space is constrained and contains unsafe states or obstacles D ⊂ Rn which need to be avoided, CCLFs provide a tool to ensure the existence of a solution x(·) ∈ S(x) of system (1) such that x(·) converges to the target (the origin) while avoiding the obstacles or unsafe states D ⊂ O for an appropriately chosen set O. While the set of unsafe states D is given, the open set O can be used as a design parameter. If D ⊂ O holds and if avoidance and convergence according to Definition 2 is satisfied for all initial values x ∈ Rn\O, then it is also ensured that the set D is never entered. An immediate application of CCLFs is obtained by considering level sets of CLFs.
Let D ⊂ Rn\{0} be an arbitrary closed set and let V be a CLF satisfying the assumptions of Theorem 1 (i.e., with respect to the unconstrained setting). Since, D is closed and 0 ∈ D/ by assumption, there exists c > 0 such that {x ∈ Rn|V(x) ≤ c} ∩ D = ∅. Thus we can define the open set O = Rn\{x ∈ Rn|V(x) ≤c}, which clearly satisfies D ⊂ O and due to the properties of a CLF, V is a CCLF with respect to the open setO. This common practice of restricting the domain of a LF or CLF to a sublevel set, and thus a forward invariant set, is quite straightforward. However, the definition of CCLFs in Definition 3 is more general and capable through nontrivial constructions, discussed in the next sections, to handle bounded open sets O.
2.3 Necessity of nonsmooth CCLFs
In the case of bounded obstacles, i.e., bounded sets Oi, i∈[1 :N], the use of nonsmooth functionsVCis essential.
This point will be made more precise in this section, before we propose a particular form of candidate CCLFs.
Lemma 1. Let Oi ⊂ Rn, i ∈ [1 : N], N ∈ N be open.
Assume thatVC is a CCLF according to Definition 3 and assume that there exists i ∈ [1 : N] such that Oi is bounded. Then there exists x∈Rn\ ∪Ni=1Oi∪ {0}
such that the gradient∇VC(x) is not defined. y The result is an immediate consequence of (Braun and Kellett, 2018, Thm. 1) which shows that a smooth function VC satisfying (4) and (5) needs to have a critical point x ∈ Rn\ ∪Ni=1Oi∪ {0}
with ∇VC(x) = 0 and thus condition (6) cannot be satisfied. The necessity of a nonsmooth function due to topological obstructions is also discussed in (Liberzon, 2003, Chapter 4) in the necessity of discontinuous feedback laws. To illustrate this problem consider the function V : Rn → R, V(x) = |x|2 as a candidate CLF and an obstacle D = {x}ˆ consisting of a single point ˆx∈Rn\B√π(0). Define the function
CD(x) = 1
2 1 + cos(|x|2xˆ)
, for|x|2xˆ≤π 0, for|x|2xˆ> π
which is continuously differentiable and has a global max- imumCD(ˆx) = 1. Withλ >0, the linear combination
VC(x) =V(x) +λCD(x), (7) satisfies the properties (4) and (5). The set O1 can be defined as an open neighborhood around ˆx. Since VC is
0 2 4
-2 -1 0 1 2
0 2 10
4 20
0 2
0 -2
Fig. 1. Visualization of the continuously differentiable function VC defined in (7) with a saddle pointx on thex1-axis. The boundary of a potential open setO1 is visualized in red.
continuously differentiable, the functionVC has a critical point inRn\(O1∪ {0}), where the gradient ofV and the gradient ofλCD cancel each other (see (Braun and Kellett, 2018, Thm. 1)). Thus condition (6) cannot be satisfied.
In Figure 1 the level sets of function (7) forλ = 20 and ˆ
x= (√
π,0)T are visualized. For the differential inclusion
˙
x∈F(x) =B|x|(0), (6) is satisfied for almost allx∈R2. However, on thex1-axis there exists a pointx∈R2where
∇VC(x) = 0 and thus a decrease cannot be obtained. This problem is closely related to the intuitive fact that on the x1-axis behind the setO1, a decision needs to be taken to avoidO1 from above or from below. In (7), the goal is to define V so that it is a CLF for the differential inclusion without obstacles, whereas CD is designed to contain the obstacle in its superlevel sets. As also observed in the literature on artificial potential fields, V and CD cannot be constructed independently and highly depend on each other to avoid the existence of local minima in the function VC. For example if the functionV(x) =|x|2is replaced by V(x) =x21+ 4x22, due to the shape of V and CD, a local minimum (and not just a saddle point) is created on the x1-axis.
3. NONSMOOTH CANDIDATE CCLFS
In this section we propose a possible form of a nonsmooth candidate CCLF. The critical condition for a function to be a CCLF is condition (6), which is discussed in Section 4 below for linear systems. For the construction we use ideas from the papers Braun et al. (2018b,c), which propose hybrid avoidance control laws for linear systems.
As in Braun et al. (2018b,c), the construction is based on avoidance points ˆx∈Rn\{0}. The open set Oaround the unsafe point then represents the obstacle that should be avoided. Here, we concentrate on a single avoidance point, (i.e., N = 1 in Definition 3) but observe that the considerations in this section carry over to the case with multiple unsafe points (obstacles).
Following Braun et al. (2018b), given the unsafe point ˆ
x∈Rn\{0}, a parameter δµ >0, and a directiond∈Rn,
|d|= 1, we define the two shifted points
cp= ˆx−pδµd, forp∈ {−1,1}. (8) The directiondwill be discussed in the subsequent section (see (20)). Consider the functions
C1(x) =−η1|x|2c1+η2, C−1(x) =−η1|x|2c−1+η2, where η1, η2 ∈R>0, and V(x) =|x|2, and combine them to form the candidate CCLF
VC(x) = max{V(x),min{C1(x), C−1(x)}}. (9) We assume thatη1 andη2 are chosen such that
VC(0) = 0 and VC(ˆx)> V(ˆx) (10)
0 1 2 -1.5
-1 -0.5 0 0.5 1 1.5
0 2
1 4
2 6
0 8
-1 0 1
Fig. 2. Visualization of the function VC defined in (9) for the parameters ˆx= (1.5,0)T,δµ= 0.4,d= (0,1)T,η1 = 5 and η2 = 7. The black lines indicate its nonsmooth domains. The boundary of a potential open setOis visualized in red.
are satisfied. This is always possible for large enough η1 and η2. Because of the “max” in (9), this guarantees that 0 is a strict global minimum and ˆx is a strict local maximum of VC. In Figure 2 an example of the function (9) with ˆx= (1.5,0)T,δµ = 0.4,d= (0,1)T, η1 = 5 and, η2 = 7 is visualized. The function in (9) is continuously differentiable almost everywhere. As compared to the continuously differentiable function visualized in Figure 1, the function in Figure 2 does not have a saddle point and thus (under the additional assumption that the right-hand side F(x) allows for it) provides the possibility to take a decision on thex1-axis using the Dini derivative as in (6).
To compute the set where the gradient does not exist, we derive in the next lemma a second representation of the functionVC defined in (9).
Lemma 2. Consider the function VC defined in (9) with η2 > 1+ηη1
1|cp|2, p ∈ {−1,1}. Additionally consider the scaled points
c∗p:= 1+ηη1
1cp, p∈ {−1,1} (11) and the radius
r∗:=q
−(1+ηη1
1)2cTpcp+1+ηη2
1. (12)
Thenr∗>0 and the functionVC can be rewritten as VC(x) =
Cp(x), if x∈ Cp∗, p∈ {−1,1}
V(x), if x∈ V, (13)
where
Cp∗={x∈Br∗(c∗p)|p(x−x)ˆ Td≥0}, p∈ {−1,1}, (14a)
V =Rn\(C∗1∪ C−1∗ ). (14b)
Proof. To establish the statements in the lemma we first compute the sets where the functionsC1,C−1and,V have the same value.
Case 1(C1(x) =C−1(x)): Since the constants cancel, we are left with the condition |x|2c1 = |x|2c−1. With the definition ofcp,p∈ {−1,1}, in (8), the left and the right- hand sides can be expanded to
(x−cp)T(x−cp) = (x−xˆ+pδµd)T(x−xˆ+pδµd)
= (x−x)ˆ T(x−x) + 2pδˆ µ(x−x)ˆ Td+ 2δ2µ from which|x|2c
1 =|x|2c−1 yields 2δµ(x−x)ˆ Td=−2δµ(x− ˆ
x)Td. Therefore, sinceδµ>0,
(x−x)ˆ Td= 0, (15) representing a hyperplane inRn orthogonal tod.
Case 2(V(x) =Cp(x) forp∈ {−1,1}): By definition, this set is characterized by
xTx=−η1(x−cp)T(x−cp) +η2
which is equivalent to
(1 +η1)xTx−2η1xTcp=η2−η1cTpcp.
Dividing by 1 +η1>0 and reordering the terms leads to xTx−2xTc∗p=1+ηη2
1 −cTpc∗p
withc∗p defined in (11). Adding |c∗p|2 to both sides we get x−c∗pT
x−c∗p
=|c∗p|2+ η2 1 +η1
−cTpc∗p (16)
=−(1+ηη1
1)2cTpcp+1+ηη2
1 = (r∗)2, which is positive by the condition on η1 and η2 and can always be achieved by selectingη2 large enough.
Note that (16) describes a sphere with centerc∗p= 1+ηη1
1cp
for p ∈ {−1,1}. Based on Lemma 2, consider the strict inequality of the right bound (10) and the representation ofVC in Lemma 2, which implies that ˆxis in the interior ofC∗−1∪ C1∗, and there existsr∈(0, r∗) such that the sets Cp={x∈Br(cp)|p(x−x)ˆ Td≥0}, (17) forp∈ {−1,1} satisfyCp⊂ Cp∗.
We may selectO= int(C1∪C−1) because for allx∈∂Owe have (usingp= 1 if (x−ˆx)Td≥0 andp=−1 otherwise)
VC(x) =Cp(x) =η2−η1r2=:c as required by (4).
Since radial unboundedness of VC follows trivially from (9) and radial unboundedness of V, we have that VC is a candidate CCLF by satisfying items (i) and (ii) of Definition 3. Since item (iii) depends on the right-hand sideF(x), in the next section we concentrate on item (iii) for a special class of differential inclusions.
4. CCLFS FOR LINEAR SYSTEMS Linear control systems
˙
x=Ax+Bu, x(0)∈Rn (18) with A ∈ Rn×n, B ∈ Rn×m represent a special class of differential inclusions (1) whereF(·) is defined by
F(x) = conv({ξ∈Rn|ξ=Ax+Bu, u∈ U(x)}) for all x∈Rn and for U(x)⊂Rm for all x. If the set of inputsU(x) is convex and compact for allx∈Rn, thenF satisfies Assumption 1.
Assume first that the linear system (18) is fully actuated, i.e.,B ∈Rn×nhas full rank. Since in this case, it is possible to move in any direction with an appropriate inputu(large enough but bounded),VC defined in (7) satisfies item (iii) of Definition 3 if it does not have local minima other than the origin. Expression (13) and the gradients ofV andCp, p∈ {−1,1} reveal that no such local minimum can exist and thusVC is a CCLF.
In a more challenging setting, consider a stabilizable lin- ear system (18) with one-dimensional input u ∈ R, i.e., B ∈ Rn×1. Additionally, assume that V(x) = xTx is a Lyapunov function for the uncontrolled system and hence that matrix AT +A is Hurwitz. This assumption is not restrictive as argued in Braun et al. (2018b), because through an appropriate coordinate transformation and by redefining the input this condition can always be achieved for a stabilizable linear system.
We derive conditions guaranteeing that the candidate
CCLF defined in (9) is a CCLF for the linear system (18) and a neighborhood Oaround a given point ˆx∈Rn. Theorem 3. Consider a stabilizable linear system (18) with one-dimensional inputu∈Rand assume thatV(x) = xTx is a Lyapunov function for the unconstrained and uncontrolled system. Let
ˆ
x∈Rn\span(B) (19)
and define the projection and the corresponding direction Pxˆ=I−xˆT1xˆxˆˆxT, d=|PPˆxB
ˆ
xB|. (20)
Then there exist parametersδµ, η1, η2∈R>0such that the control law
u(x) =−hx−c
∗ p,Axi
hx−c∗p,Bi (21)
is well-defined for all x ∈ V ∩ Cp∗, p ∈ {−1,1}. If additionally
0> max
x∈V∩Cp∗
dVC
x;−hx−c
∗ p,Axi hx−c∗p,Bi
(22) for p ∈ {−1,1} then VC defined in Lemma (13) is a CCLF according to Definition 3 for O ⊂ Rn defined as
a neighborhood around ˆx. y
The direction ddefines the orientation of the hyperplane (15) and, by construction, ˆxTd = 0 which simplifies the representation of (15) to {x∈Rn|xTd= 0} and similarly the expressions of Cp, C∗p, p∈ {−1,1}, in (14) and (17).
Moreover, d is aligned with the projection of B on the hyperplanePxˆxfor x∈Rn. For suitable non-controllable (but stabilizable) systems, it can be shown that condition (19) is necessary for the existence of a CCLF. Condition (22) guarantees that VC decreases along trajectories in V ∩ Cp∗, p ∈ {−1,1}, if the control law (21) is used. A more explicit condition for (22) is given in Remark 1 after the proof of Theorem 3. A simplification of this condition is a goal of future research.
Proof. As a first step, we show that the control law (21) is well-defined and provide a motivation for the selection of u. For the sphere S∗ :={x∈ Rn| |x|c∗p =r∗}, Braun et al. (2018b) proposed the control law (21) based on the condition dtd|x(t)|2c∗
p = 0. For x(0) ∈ S∗, by construction, the inputuin (21) ensures thatx(t)∈ S∗ for allt≥0, if hx−c∗p, bi 6= 0 is satisfied.
For (21) to be well-defined, it is sufficient ifhx−c∗p, bi 6= 0 holds for all x ∈ Cp∗, which can be achieved by making the intersection of the ball Br∗(c∗p) and the half-space p·xTd≥0 small. To this end we define
ˆ
x∗=12 c∗1+c∗−1
= 121+ηη1
1(c1+c−1) =1+ηη1
1x.ˆ which is a linear combination of ˆxand ˆx∗ ∈/ span(B). In the same way asδµdefines the distance fromcpto ˆx(and thus to the hyperplanexTd= 0), we can define
δµ∗=|ˆx∗−c∗p|=
η1
1+η1
δµ (23)
as the distance fromc∗p to the hyperplanexTd= 0. Then, as visualized in Figure 3, for (21) to be well-defined inCp∗, p∈ {−1,1}, it is sufficient that
δ∗µ r∗ >
hˆx,Bi
|ˆx||B|
(24)
0
α α β α
γ B d
ˆ x∗ δˆ∗µ
r∗
Fig. 3. Visualization of the relation between the angleγ and the angleα. To ensure that hx−c∗p, Bi 6= 0 for all x ∈ C∗p it is sufficient thatα < γholds.
be satisfied. To see this, observe thatα:= arccos δ∗µ/r∗ defines the maximal angle of a tangent vector of x ∈ Cp∗ andx−c∗p, and
γ:= arccos
hˆx∗,Bi
|ˆx∗||B|
= arccos
hˆx,Bi
|ˆx||B|
defines the angle between ˆx and B. Thus α < γ is equivalent to condition (24) and ensures thatCp∗ does not contain pointsxsuch thatx−c∗pandBare perpendicular.
Note thatr∗ can be made arbitrarily small by increasing η2 for fixed η1. This implies that r∗ −δµ∗ > 0 can be made arbitrarily small and therefore (24) can be achieved through an appropriate selection ofδµ,η1andη2.
In Section 3 we have shown that item (i) and (ii) of Defin- tion 3 are satisfied forVC defined in (9) and appropriate parameters η1, η2 and here we have shown that addition- ally condition (24) can be satisfied. Thus, we concentrate on the decrease condition (6) in item (iii) of Definition 3 in the following. For the proof, we use the representation of VC in (13) and in particular the definition of the sets (14) to verify (6) for allx∈Rn.
Case 1(x∈ V): SinceAT +Ais Hurwitz by assumption, the inputu= 0 satisfiesdVC(x, Ax) =xT(AT +A)x <0 for all x ∈ V. This implies that an appropriate function ρ∈ Pcan be defined.
Case 2(x∈ V ∩ Cp∗,p∈ {−1,1}): If condition (24) is sat- isfied, the control law (21) is well-defined onCp∗ and (22) ensures a decrease.
Case 3(x∈ Cp∗\(V ∪ C−p∗ ),p∈ {−1,1}): Due to the selec- tion ofr∈(0, r∗) to defineCpin (17) and due the definition ofδµ? in (23) and property (24), the estimate
δµ r > δ
∗ µ
r∗ >
hˆx,Bi
|ˆx||B|
(25) holds. Similar to control law (21), we define
u(x) =1−hx−chx−cp,Axi
p,Bi . (26)
based on the condition dtd|x(t)|2cp= 2,i.e.,u(x) is defined such that the distance tocpincreases, and thusVC(x(t)) = Cp(x(t)) decreases along trajectoriesx(t)∈ Cp∗\(V ∪ C−p∗ ).
Inequality (25) ensures that the control law is well-defined for allx(t)∈ Cp∗.
Case 4(x∈ C−1∗ ∩ C∗1): On C∗−1∩ C1∗, the function VC is concave (and thus, in particular, semiconcave, see (Clarke, 2011, Sec. 5)) as the minimum of two concave functions C−1(x) and C1(x). Thus, the control law (26) for p ∈ {1,−1} arbitrary ensures that (6) is satisfied on (C−1∗ ∩ C1∗)\xˆwhich concludes the proof. 2
